Abstract

A theoretical analysis of the effect of recording nonlinearities upon the image reconstructed from a hologram made of a diffuse object is presented. Extensive experimental evidence which supports this theory is also given. In particular, it is shown that the magnitude of the nonlinearity noise can be calculated knowing the shape of the amplitude transmittance–exposure (TaE) curve, the bias transmittance (Tb), and the ratio of the reference beam to the object beam intensity (K). It is further shown that for diffuse objects the shape of the nonlinearity noise distribution can be calculated from the shape of the object.

© 1970 Optical Society of America

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References

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  1. A. Kozma, J. Opt. Soc. Amer. 56, 428 (1966).
    [CrossRef]
  2. A. A. Friesem, J. S. Zelenka, Appl. Opt. 6, 1755 (1967).
    [CrossRef] [PubMed]
  3. J. M. J. Tokarski, Appl. Opt. 7, 989 (1968).
    [CrossRef] [PubMed]
  4. J. W. Goodman, G. R. Knight, J. Opt. Soc. Amer. 58, 1276 (1968).
    [CrossRef]
  5. O. Bryngdal, A. Lohmann, J. Opt. Soc. Amer. 58, 1325 (1968).
    [CrossRef]
  6. A. Kozma, Opt. Acta 15, 527 (1968).
    [CrossRef]
  7. D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Company, New York, 1960), p. 356.
  8. Ref. 7, p. 402.
  9. A. Kozma, Ph.D. Thesis, University of London, 1968.
  10. J. W. Goodman, Proc. IEEE 53, 1688 (1965).
    [CrossRef]
  11. The normalized Ta− Ecurve is obtained from the ordinary Ta− Ecurve by shifting and scaling the exposure coordinate so that a normalized exposure E= 0 produces a desired bias transmittance Tband a normalized exposure E= −1 produces unity transmittance.
  12. The area of the points was large enough to leave unaffected the distribution of the intensities in the spectrum of the randomly-scattering plate which backed the transparency.
  13. C. B. Burckhardt, Appl. Opt. 6, 1359 (1967).
    [CrossRef] [PubMed]

1968 (4)

J. M. J. Tokarski, Appl. Opt. 7, 989 (1968).
[CrossRef] [PubMed]

J. W. Goodman, G. R. Knight, J. Opt. Soc. Amer. 58, 1276 (1968).
[CrossRef]

O. Bryngdal, A. Lohmann, J. Opt. Soc. Amer. 58, 1325 (1968).
[CrossRef]

A. Kozma, Opt. Acta 15, 527 (1968).
[CrossRef]

1967 (2)

1966 (1)

A. Kozma, J. Opt. Soc. Amer. 56, 428 (1966).
[CrossRef]

1965 (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Bryngdal, O.

O. Bryngdal, A. Lohmann, J. Opt. Soc. Amer. 58, 1325 (1968).
[CrossRef]

Burckhardt, C. B.

Friesem, A. A.

Goodman, J. W.

J. W. Goodman, G. R. Knight, J. Opt. Soc. Amer. 58, 1276 (1968).
[CrossRef]

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Knight, G. R.

J. W. Goodman, G. R. Knight, J. Opt. Soc. Amer. 58, 1276 (1968).
[CrossRef]

Kozma, A.

A. Kozma, Opt. Acta 15, 527 (1968).
[CrossRef]

A. Kozma, J. Opt. Soc. Amer. 56, 428 (1966).
[CrossRef]

A. Kozma, Ph.D. Thesis, University of London, 1968.

Lohmann, A.

O. Bryngdal, A. Lohmann, J. Opt. Soc. Amer. 58, 1325 (1968).
[CrossRef]

Middleton, D.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Company, New York, 1960), p. 356.

Tokarski, J. M. J.

Zelenka, J. S.

Appl. Opt. (3)

J. Opt. Soc. Amer. (3)

A. Kozma, J. Opt. Soc. Amer. 56, 428 (1966).
[CrossRef]

J. W. Goodman, G. R. Knight, J. Opt. Soc. Amer. 58, 1276 (1968).
[CrossRef]

O. Bryngdal, A. Lohmann, J. Opt. Soc. Amer. 58, 1325 (1968).
[CrossRef]

Opt. Acta (1)

A. Kozma, Opt. Acta 15, 527 (1968).
[CrossRef]

Proc. IEEE (1)

J. W. Goodman, Proc. IEEE 53, 1688 (1965).
[CrossRef]

Other (5)

The normalized Ta− Ecurve is obtained from the ordinary Ta− Ecurve by shifting and scaling the exposure coordinate so that a normalized exposure E= 0 produces a desired bias transmittance Tband a normalized exposure E= −1 produces unity transmittance.

The area of the points was large enough to leave unaffected the distribution of the intensities in the spectrum of the randomly-scattering plate which backed the transparency.

D. Middleton, Introduction to Statistical Communication Theory (McGraw-Hill Book Company, New York, 1960), p. 356.

Ref. 7, p. 402.

A. Kozma, Ph.D. Thesis, University of London, 1968.

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Figures (16)

Fig. 1
Fig. 1

Bar pattern object. (a) Shape of pattern. (b) Intensity autocorrelation function along a central horizontal section, and along two vertical sections through (A) and (B).

Fig. 2
Fig. 2

Calculated image intensity levels for bar pattern object. Object shape-dependent noise (1); object intensity-dependent noise (2); and signal (3).

Fig. 3
Fig. 3

Regular 16 × 16 array object, with points 9 and 21 missing.

Fig. 4
Fig. 4

Hologram recording system for nonlinearity noise study. The objects (2.5 cm × 2.5 cm) were located 53 cm in front of hologram recording plane (M—mirror, BS—beamsplitter).

Fig. 5
Fig. 5

Normalized T a E curves: (a) measured and used in present study; (b) reported earlier.6 (649F plate, developed in D-19 for 5 min.) E is the normalized exposure, T a (E) = T b + F(E), where T b is the bias transmittance (in this case 0.5). Crosses on curve (a) indicate points of experimental fit to the polynomial: F ( E ) = υ = 1 9 s υ E υ , where s1 = −0.65742, s2 = 0.24179, s3 = 0.15012, s4 = −0.17341, s5 = 0.069183, s6 = −0.014483, s7 = 0.0016806, s8 = −0.000-10149, and s9 = 0.0000024587.

Fig. 6
Fig. 6

Calculated values of the noise coefficients α and β, where α = β 1 2 (α determines the magnitude of the object intensity dependent noise).

Fig. 7
Fig. 7

Calculated values of the noise coefficient γ (γ determines the magnitude of the first-order object shape-dependent noise).

Fig. 8
Fig. 8

Calculated values of the noise coefficient δ (δ determines the magnitude of the second-order object shape-dependent noise).

Fig. 9
Fig. 9

Reconstructed images of the square transparency, illustrating object shape-dependent noise (K = 1 hologram). The thresholds are determined with reference to the exposure for which the image was just visible, and are expressed as decibels (e.g., doubling the exposure time drops the threshold of visibility by 3 dB for the high contrast film used).

Fig. 10
Fig. 10

Photometric scans across the square transparency images. Holograms made with (a) K = 1.8, T b = 0.35; and (b) K = 9.7, T b = 0.64. I N , calculated nonlinearity noise levels (— — —); I B , observed background noise level (…); observed intensity levels (—); (I N + I B ) (○—○—○); and calculated (I S + I N + I B ) (—··).

Fig. 11
Fig. 11

Photometric scan across the step wedge image. Hologram made with K = 33, T b = 0.49. The five steps of the object intensity are indicated I1I5, each with width of 0.2 L. I N , calculated nonlinearity noise level (— — —); I B , observed background noise level (…); observed intensity levels (—); calculated signal and noise intensity levels (—··).

Fig. 12
Fig. 12

Photometric scans across the step wedge images. Holograms made with (a) K = 3.8, T b = 0.47; and (b) K = 3.8, T b = 0.72. (Captions on Figs. 10 and 11 apply.)

Fig. 13
Fig. 13

Characteristics of a regular half-filled, 16 × 16 array. (a) Image of the array (hologram made with K = 29), showing central cross section scanned. (b) Image of the array (hologram made with K = 1.0, T b = 0.35), illustrating object shape-dependent noise. (c) Intensity autocorrelation function.

Fig. 14
Fig. 14

Photometric scans across central cross sections of array images. Holograms made with (a) K = 1, T b = 0.6; and (b) K = 4, T b = 0.7. Filled array points (—); missing array points (—). I N 1 and I N 2, calculated nonlinearity noise levels at center and edge of array. I B , background noise between array points.

Fig. 15
Fig. 15

Comparison of observed with predicted noise coefficient, γ. (a) Predictions for T b = 0.3 and 0.6 compared with data taken with 0.2 ≤ T b ≤ 0.4 (×) and 0.55 ≤ T b ≤ 0.65 (●). Observed values of γ for which the background noise I B , was allowed for (□). (b) Predictions for T b = 0.5 and 0.7 compared with data taken with 0.45 < T b ≤ 0.55 (●) and 0.65 < T b ≤ 0.75 (×).

Fig. 16
Fig. 16

Comparison of predicted values of noise coefficient, γ. γ c represents the first eight terms in the expansion of γ. γ′ = (C11/C10)2 is the first term in the expansion of γ (valid for both −1 < E < 7 and −1 < E < 3). γ1′ = (C11′/C10′)2 is the first term in the expansion for γ appropriate for the T a E curve of Fig. 5(b).6

Equations (49)

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a ( x ) e j θ ( x ) = j λ z A 0 0 ( u ) e j k R ( x , u ) d , u ,
E { e j [ Ψ ( u 1 ) Ψ ( u 2 ) ] } = 1 ; u 1 = u 2 , = 0 ; otherwise ,
E [ a n ( x 1 ) a m ( x 2 ) ] = ( 2 σ 2 ) ( n + m ) / 2 Γ ( n + 2 2 ) Γ ( m + 2 2 ) × F ( n 2 , m 2 , 1 ; | ϕ | 2 4 σ 4 ) ,
ϕ ( x 1 , x 2 ) = E { a ( x 1 ) a ( x 2 ) exp j [ θ ( x 1 ) θ ( x 2 ) ] }
ϕ ( x 1 , x 2 ) = exp j ( k / 2 z ) [ ( x 1 2 x 2 2 ) + ( y 1 2 y 2 2 ) ] ( λ z ) 2 A 0 d u 1 × A 0 d u 2 a 0 ( u 1 ) a 0 * ( u 2 ) E { exp j [ Ψ ( u 1 ) Ψ ( u 2 ) ] } × exp j ( k / z ) ( x 1 u 1 x 2 u 2 + y 1 υ 1 y 2 υ 2 ) × exp j ( k / 2 z ) ( u 1 2 u 2 2 + υ 1 2 υ 2 2 ) .
| ϕ ( x 1 , x 2 ) | 2 = | A 0 ( λ z ) 2 A 0 d u | a 0 ( u ) | 2 × exp j ( k / z ) [ ( x 1 x 2 ) u + ( y 1 y 2 ) υ ] | 2 .
T ( x ) = [ 2 ( K ) 1 2 / ( K + 1 ) ] s 1 [ 1 + B ( x ) ] 1 / 2 cos [ k R r θ ( x ) ] ,
T d ( x ) = 2 ( K ) 1 2 ( K + 1 ) s 1 C 10 [ n = 0 C 1 n B n ( x ) C 10 ] × [ 1 + B ( x ) ] 1 / 2 cos [ k R r θ ( x ) ] .
B ( x ) = { a 2 ( x ) / E [ a 2 ( x ) ] } 1.
U d ( x ) = a r D { n = 0 C 1 n C 10 B n ( x ) } a ( x ) ( 2 ) 1 2 σ e j θ ( x ) ,
D = ( K ) 1 2 s 1 C 10 / ( K + 1 ) .
S ( x ) = a r [ D / ( 2 ) 1 2 σ ] a ( x ) e j θ ( x ) ,
N ( x ) = [ n = 1 C 1 n C 10 B n ( x ) ] D a r a ( x ) ( 2 ) 1 2 σ e j θ ( x ) ,
U d ( x ) = D a r ( 2 ) 1 2 σ { j λ z e j k z A 0 a 0 ( u ) e j Ψ ( u ) e j k R ( x ; u ) d u + [ n = 1 C 1 n C 10 B n ( x ) ] j λ z e j k z A 0 a 0 ( u ) e j Ψ ( u ) e j k R ( x ; u ) d u } .
U ( u i ) = j λ z e j k z A t U d ( x ) e j k R ( x ; u i ) d x ,
U ( u i ) = D a r ( λ z ) 2 ( 2 ) 1 2 σ [ S υ ( u i ) + N υ ( u i ) ] ,
S υ ( u i ) = A 0 d u A t d x a 0 ( u ) e j Ψ ( u ) × exp [ j ( k / 2 z ) ( u 2 u i 2 + υ 2 υ i 2 ) ] × exp { j ( k / z ) [ x ( u u i ) + y ( υ υ i ) ] }
N υ ( u i ) = A 0 d u A t d x a 0 ( u ) e j Ψ ( u ) n = 1 C 1 n C 10 B n ( x ) × exp [ j ( k / 2 z ) ( u 2 u i 2 + υ 2 υ i 2 ) ] × exp { j ( k / z ) [ x ( u u i ) + y ( υ υ i ) ] } .
( I ) i = D 2 I r ( λ z ) 4 2 σ 3 { E [ S υ ( u i ) S υ * ( u i ) ] + E [ S υ ( u i ) N υ * ( u i ) ] + E [ S υ * ( u i ) N υ ( u i ) ] + E [ N υ ( u i ) N υ * ( u i ) ] }
( I ) i = D 2 I r ( λ z ) 4 2 σ 2 [ I ( u i ) + I n ( u i ) + I n ( u i ) ] ,
I = E ( S υ S υ * ) ,
I n = E ( S υ N υ * ) + E ( S υ * N υ ) ,
I n = E ( N υ N υ * ) .
I ( u i ) = A 0 d u A 0 d u A t d x A t d x a 0 ( u ) a 0 * ( u ) × E { exp j [ Ψ ( u ) Ψ ( u ) ] } exp [ j ( k / 2 z ) ( u 2 u 2 + υ 2 υ 2 ) ] × exp { j ( k / z ) [ x ( u u i ) + y ( υ υ i ) x ( u u i ) y ( υ υ i ) ] } .
I ( u i ) = A 0 A 0 d u A t d x A 0 d x | a 0 ( u ) | 2 × exp { j ( k / z ) [ ( x x ) ( u u i ) + ( y y ) ( υ υ i ) ] } .
I ( u i ) = A 0 ( L x L y ) 2 A 0 d u | a 0 ( u ) | 2 sin c 2 k L x 2 z ( u u i ) × sin c 2 k L y 2 z ( υ υ i ) ,
I ( u i ) = A 0 ( L x L y ) 2 | a 0 ( u i ) | 2 * h ( u i ) ,
I n ( u ) = A 0 d u A 0 d u A t d x A t d x × a 0 ( u ) a 0 * ( u ) E { exp j [ Ψ ( u ) Ψ ( u ) ] } E [ n = 1 C 1 n C 10 B n ( x ) ] × exp [ j ( k / 2 z ) ( u 2 u 2 + υ 2 υ 2 ) ] exp { j ( k / z ) [ x ( u u i ) + y ( υ υ i ) x ( u u i ) y ( υ υ i ) ] } + c . c . ,
I n ( u i ) = A 0 A t d x A t d x A 0 d u | a 0 ( u ) | 2 × E [ n = 1 C 1 n C 10 B n ( x ) ] exp { j ( k / z ) [ ( x x ) ( u u i ) + ( y y ) ( υ υ i ) ] } + c . c .
I n ( u i ) = 2 A 0 ( L x L y ) 2 α A 0 d u | a 0 ( u ) | 2 h ( u u i , υ υ i )
I n ( u i ) = 2 A 0 ( L x L y ) 2 α | a 0 ( u i ) | 2 * h ( u i ) ,
α = n = 1 p = 0 ( 1 ) n + p n ! ( n p ) ! C 1 n C 10 .
I n ( u i ) = A 0 d u A 0 d u A t d x A t d x a 0 ( u ) × a 0 * ( u ) E ( exp { j [ Ψ ( u ) Ψ ( u ) ] } ) E [ n = 1 m = 1 C 1 n C 1 m ( C 10 ) 2 × B n ( x ) B m ( x ) ] exp [ j ( k / 2 z ) ( u 2 u 2 + υ 2 υ 2 ) ] × exp { j ( k / z ) [ x ( u u i ) + y ( υ υ i ) x ( u u i ) y ( υ υ i ) ] } .
I n ( u i ) = A 0 A 0 d u | a 0 ( u ) | 2 A t d x A t d x × E [ n = 1 m = 1 C 1 n C 1 m ( C 10 ) 2 B n ( x ) B m ( x ) ] × exp { j ( k / z ) [ ( x x ) ( u i u ) + ( y y ) ( υ i υ ) ] } .
E [ n = 1 m = 1 C 1 n C 1 m ( C 10 ) 2 B n ( x ) B m ( x ) ] = [ n = 1 m = 1 C 1 n C 1 m ( C 10 ) 2 ( a 2 ( x ) 2 σ 2 1 ) n ( a 2 ( x ) 2 σ 2 1 ) m ] .
E [ n = 1 m = 1 C 1 n C 1 m ( C 10 ) 2 B n ( x ) B m ( x ) ] = β + γ | ϕ ( x , x ) | 2 4 σ 4 + δ ( | ϕ ( x , x ) | 2 4 σ 4 ) 2 + ,
β = ( n = 1 p = 0 n C 1 n C 10 ( 1 ) n + p n ! ( n p ) ! ) 2 ,
γ = ( n = 1 p = 0 n C 1 n C 10 ( 1 ) n + p n ! ( n p ) ! ( p ) ) 2 ,
δ = ( n = 1 p = 0 n C 1 n C 10 ( 1 ) n + p n ! ( n p ) ! ( p ) ( p + 1 ) 2 ) 2 .
I n ( u i ) = A 0 A 0 d u | a 0 ( u ) | 2 A t d x A t d x × { β + γ | ϕ ( x , x ) | 2 4 σ 4 + δ [ | ϕ ( x , x ) | 2 4 σ 4 ] 2 + } × exp { j ( k / z ) [ ( x x ) ( u u i ) + ( y y ) ( υ υ i ) ] } .
I n ( u i ) = A 0 ( L x L y ) 2 [ β A 0 d u | a 0 ( u ) | 2 h ( u u i , υ υ i ) + γ C A 0 d u | a 0 ( u ) | 2 | a 0 ( u u i , υ υ i ) | 2 | a 0 ( u u i , υ υ i ) | 2 * h ( u u i , υ υ i ) + δ C 2 A 0 d u | a 0 ( u ) | 2 | a 0 ( u u i , υ υ i ) | 2 | a 0 ( u u i , υ υ i ) | 2 * | a 0 ( u u i , υ υ i ) | 2 | a 0 ( u u i , υ υ i ) | 2 * h ( u u i , υ υ i ) + ] ,
C = [ A 0 d u | a 0 ( u ) | 2 ] 2 .
I n ( u i ) = A 0 ( L x L y ) 2 γ C A 0 d u | a 0 ( u ) | 2 | a 0 ( u u i , υ υ i ) | 2 | a 0 ( u u i , υ υ i ) | 2 * h ( u u i , υ υ i ) .
( I ) i = D 2 I r ( L x L y ) 2 A 0 2 σ 2 ( λ z ) 4 { | a 0 ( u i ) | 2 + 2 α | a 0 ( u i ) | 2 + γ a ˆ ( u i ) } * h ( u i ) .
a ˆ ( u i ) = 1 C [ A 0 d u | a 0 ( u ) | 2 | a 0 ( u u i , υ υ i ) | 2 | a 0 ( u u i , υ υ i ) | 2 ] .
S / N = | a 0 ( u i ) | 2 * h ( u i ) [ 2 α | a 0 ( u i ) | 2 + γ a ˆ ( u i ) ] * h ( u i ) .
I n ( u i ) = 0.022 a 2 sin c 2 [ π L x λ z ( u i + L u 10 ) ] sin c 2 [ π L y λ z ( υ i L υ 10 ) ]
T b = T b m [ 2 K / ( K + 1 ) 2 ] s 2 T 0 ,
T 0 = [ 1 + 3 K ( K + 1 ) 2 s 4 s 2 + 10 K 2 ( K + 1 ) 4 s 6 s 2 + 35 K 3 s 8 ( K + 1 ) 6 s 2 + ] .

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